In-Class Exercise 07

In-Class Exercise
R
spdep

Geographically Weighted Regression

Author

Khant Min Naing

Published

March 11, 2024

Modified

March 11, 2024

Geographical Weighted Regression

1.0 Overview

Geographically weighted regression (GWR) is a spatial statistical technique that takes non-stationary variables into consideration (e.g., climate; demographic factors; physical environment characteristics) and models the local relationships between these independent variables and an outcome of interest (also known as dependent variable). In this hands-on exercise, we explore how to build hedonic pricing models by using GWR methods. The dependent variable is the resale prices of condominium in 2015. The independent variables are divided into either structural and locational.

2.0 Importing Datasets and Packages

Firstly, we will install and import necessary R-packages for this modelling exercise. The R packages needed for this exercise are as follows:

  • R package for building OLS and performing diagnostics tests

Reflection

oslrr is a good package that compiles multiple important functions for model diagnostics. However, the package is specifically prepared for OLS models, so it cannot be applied to other regression models such as simple linear regression, multiple linear regression or binary logistic regression.

  • R package for calibrating geographical weighted family of models

  • R package for multivariate data visualisation and analysis

  • Spatial data handling

    • sf

  • Attribute data handling

    • tidyverse, especially readr, ggplot2 and dplyr

  • Choropleth mapping

    • tmap

  • Creating publication ready HTML tables

    • vtable, tableHTML
pacman::p_load(olsrr, corrplot, ggpubr, sf, ggstatsplot, spdep, GWmodel, tmap, tidyverse, gtsummary,vtable, sjPlot, sjmisc, sjlabelled, tableHTML)

Next, two data sets will be used in this model building exercise, they are:

  • URA Master Plan subzone boundary in shapefile format (i.e. MP14_SUBZONE_WEB_PL)

  • condo_resale_2015 in csv format (i.e. condo_resale_2015.csv)

mpsz = st_read(dsn = "~/IS415-GAA/data/geospatial", layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/khantminnaing/IS415-GAA/data/geospatial' using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21
condo_resale = read_csv("~/IS415-GAA/data/aspatial/Condo_resale_2015.csv")

3.0 Data Wrangling

3.1 Geospatial Data Wrangling

We use st_transform() to update the imported mpsz with the correct ESPG code (i.e. 3414). Then, we use st_bbox() to view the extent of mpsz_svy21.

mpsz_svy21 <- st_transform(mpsz, 3414)
st_crs(mpsz_svy21)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]
st_bbox(mpsz_svy21)
     xmin      ymin      xmax      ymax 
 2667.538 15748.721 56396.440 50256.334

3.1 Aspatial Data Wrangling

We use glimpse() to have a quick overview of the data structure of condo_resale data.

glimpse(condo_resale)
Rows: 1,436
Columns: 23
$ LATITUDE             <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE            <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE             <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE        <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM             <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE                  <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD             <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE       <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE     <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET   <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN    <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT             <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK            <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH     <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL   <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET     <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP        <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS          <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY      <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD             <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR       <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…

To calculate the summary statistics of condo_resale data frame, we use st().

st(condo_resale)
Summary Statistics
Variable N Mean Std. Dev. Min Pctl. 25 Pctl. 75 Max
LATITUDE 1436 1.3 0.038 1.2 1.3 1.4 1.5
LONGITUDE 1436 104 0.067 104 104 104 104
POSTCODE 1436 440439 201080 18965 259849 589486 828833
SELLING_PRICE 1436 1751211 1272778 540000 1100000 1950000 18000000
AREA_SQM 1436 137 58 34 103 156 619
AGE 1436 12 8.6 0 5 18 37
PROX_CBD 1436 9.3 4.3 0.39 5.6 13 19
PROX_CHILDCARE 1436 0.33 0.33 0.0049 0.17 0.37 3.5
PROX_ELDERLYCARE 1436 1.1 0.62 0.055 0.61 1.4 3.9
PROX_URA_GROWTH_AREA 1436 4.6 2 0.21 3.2 5.8 9.2
PROX_HAWKER_MARKET 1436 1.3 1 0.052 0.55 1.7 5.4
PROX_KINDERGARTEN 1436 0.46 0.26 0.0049 0.28 0.58 2.2
PROX_MRT 1436 0.67 0.48 0.053 0.35 0.85 3.5
PROX_PARK 1436 0.5 0.33 0.029 0.26 0.66 2.2
PROX_PRIMARY_SCH 1436 0.75 0.49 0.077 0.44 0.95 3.9
PROX_TOP_PRIMARY_SCH 1436 2.3 1.4 0.077 1.3 2.9 6.7
PROX_SHOPPING_MALL 1436 1 0.66 0 0.53 1.4 3.5
PROX_SUPERMARKET 1436 0.61 0.33 0 0.37 0.79 2.2
PROX_BUS_STOP 1436 0.19 0.25 0.0016 0.098 0.22 2.5
NO_Of_UNITS 1436 409 273 18 189 590 1703
FAMILY_FRIENDLY 1436 0.49 0.5 0 0 1 1
FREEHOLD 1436 0.42 0.49 0 0 1 1
LEASEHOLD_99YR 1436 0.49 0.5 0 0 1 1

Finally, we will convert this aspatial data frame into a sf object. To do so, we will use st_as_sf() of sf package.

condo_resale.sf <- st_as_sf(condo_resale,
                            coords = c("LONGITUDE", "LATITUDE"),
                            crs=4326) %>% st_transform(crs=3414)
head(condo_resale.sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
     <dbl>         <dbl>    <dbl> <dbl>    <dbl>          <dbl>            <dbl>
1   118635       3000000      309    30     7.94          0.166            2.52 
2   288420       3880000      290    32     6.61          0.280            1.93 
3   267833       3325000      248    33     6.90          0.429            0.502
4   258380       4250000      127     7     4.04          0.395            1.99 
5   467169       1400000      145    28    11.8           0.119            1.12 
6   466472       1320000      139    22    10.3           0.125            0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>

4.0 Exploratory Data Analysis (EDA)

4.1 EDA Using Statistical Graphics

We can plot the distribution of different data columns by using appropriate Exploratory Data Analysis (EDA). As an example, we will plot SELLING_PRICE.

ggplot(data=condo_resale.sf, aes(x=`SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="#e9531e")

From the figure above, it seems like there is a right skewed distribution. This means that more condominium units were transacted at relative lower prices.

Statistically, the skewed dsitribution can be normalised by using log transformation. The code chunk below is used to derive a new variable called LOG_SELLING_PRICE by using a log transformation on the variable SELLING_PRICE. It is performed using mutate() of dplyr package.

condo_resale.sf <- condo_resale.sf %>%
  mutate(`LOG_SELLING_PRICE` = log(SELLING_PRICE))

ggplot(data=condo_resale.sf, aes(x=`LOG_SELLING_PRICE`)) +
  geom_histogram(bins=20, color="black", fill="#e9531e")

4.2 EDA Using Multiple Histogram Plots Distribution of Variables

In previous section, we specify a varible to plot. In this section, we will instead draw a small multiple histograms (also known as trellis plot) by using ggarrange() of ggpubr package. In this way, we can see the distribution plots of different variables at the same time.

AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) + 
  geom_histogram(bins=20, color="black", fill="#e9531e")

AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
  geom_histogram(bins=20, color="black", fill="#e9531e")

PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
  geom_histogram(bins=20, color="black", fill="#e9531e")

PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) + 
  geom_histogram(bins=20, color="black", fill="#DC375E")

PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
  geom_histogram(bins=20, color="black", fill="#DC375E")

PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_URA_GROWTH_AREA`)) +
  geom_histogram(bins=20, color="black", fill="#DC375E")

PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
  geom_histogram(bins=20, color="black", fill="#AE4285")

PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
  geom_histogram(bins=20, color="black", fill="#AE4285")

PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
  geom_histogram(bins=20, color="black", fill="#AE4285")

PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
  geom_histogram(bins=20, color="black", fill="#71508F")

PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="#71508F")

PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf, 
                               aes(x= `PROX_TOP_PRIMARY_SCH`)) +
  geom_histogram(bins=20, color="black", fill="#71508F")

ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, 
          PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
          PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,  
          ncol = 3, nrow = 4)

4.3 Drawing Statistical Point Map

Next, we will learn how to reveal the geospatial distribution condominium resale prices in Singapore using statistical point maps. To plot such maps, we will prepare using tmap package.

  • First, we will turn on the interactive mode of tmap by using the code chunk below.

  • Then, we will create an interactive point symbol map using the data values from SELLING_PRICE column.

  • Next, we will turn R display into plot mode.

#tmap_mode("view")
tmap_options(check.and.fix = TRUE)
tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.sf) +  
  tm_dots(col = "SELLING_PRICE",
          palette = "plasma",
          alpha = 1,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))

5.0 Hedonic Pricing Modelling in R

In this section, we will explore how to build a hedonic pricing model for condominium resale units using lm() of R.

5.1 Simple Linear Regression Method

First, we will build a simple linear regression model by using SELLING_PRICE as the dependent variable and AREA_SQM as the independent variable.

condo.slr <- lm(formula=SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)

lm() returns an object of class “lm” or for multiple responses of class c(“mlm”, “lm”).

The functions summary() and anova() can be used to obtain and print a summary and analysis of variance table of the results. The generic accessor functions coefficients, effects, fitted.values and residuals extract various useful features of the value returned by lm.

tab_model(condo.slr)
  Dependent variable
Predictors Estimates CI p
(Intercept) -258121.06 -382717.70 – -133524.43 <0.001
AREA SQM 14719.03 13879.23 – 15558.83 <0.001
Observations 1436
R2 / R2 adjusted 0.452 / 0.451

The output report reveals that the SELLING_PRICE can be explained by using the formula:

\(y = -258121.1 + 14719x1\)

The R-squared of 0.4518 reveals that the simple regression model built is able to explain about 45% of the resale prices.

Since p-value is much smaller than 0.001, we will reject the null hypothesis that mean is a good estimator of SELLING_PRICE. This will allow us to infer that simple linear regression model above is a good estimator of SELLING_PRICE.

The Coefficients: section of the report reveals that the p-values of both the estimates of the Intercept and ARA_SQM are smaller than 0.001. In view of this, the null hypothesis of the B0 and B1 are equal to 0 will be rejected. As a results, we will be able to infer that the B0 and B1 are good parameter estimates.

To visualise the best fit curve on a scatterplot, we can incorporate lm() as a method function in ggplot’s geometry as shown in the code chunk below.

ggplot(data=condo_resale.sf,  
       aes(x=`AREA_SQM`, y=`SELLING_PRICE`)) +
  geom_point() +
  geom_smooth(method = lm)

Figure above reveals that there are a few statistical outliers with relatively high selling prices.

5.2 Multiple Linear Regression Method

Before building a multiple regression model, it is important to ensure that the indepdent variables used are not highly correlated to each other. If these highly correlated independent variables are used in building a regression model by mistake, the quality of the model will be compromised. This phenomenon is known as multicollinearity in statistics.

Correlation matrix is commonly used to visualise the relationships between the independent variables. In this section, the corrplot package will be used to display the correlation matrix of the independent variables in condo_resale data frame.

corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         col=colorRampPalette(c("#E9531E","#F4E8EC","#B445B8"))(10),
         tl.pos = "td", tl.cex = 0.5,tl.col = "black", number.cex = 0.5, method = "number", type = "upper")

corrplot(cor(condo_resale[, 5:23]), diag = FALSE, order = "AOE",
         col=colorRampPalette(c("#E9531E","#F4E8EC","#B445B8"))(10),
         tl.pos = "td", tl.cex = 0.5,tl.col = "black", number.cex = 0.5, method = "ellipse", type = "upper")

Matrix reorder is very important for mining the hiden structure and patter in the matrix. There are four methods in corrplot (parameter order), named “AOE”, “FPC”, “hclust”, “alphabet”. In the code chunk above, AOE order is used. It orders the variables by using the angular order of the eigenvectors method suggested by Michael Friendly.

From the scatterplot matrix, it is clear that Freehold is highly correlated to LEASE_99YEAR. In view of this, it is wiser to only include either one of them in the subsequent model building. As a result, LEASE_99YEAR is excluded in the subsequent model building.

Another way to visualise more sophisticated correlation matrix is to use ggstatsplot package. Particularly, ggcormat() function can be used to create the correlation matrix with details from statistical tests included in the plots themselves. The resulting plot provides a visual and statistical summary of the relationships between the variables in the selected columns of the data frame. It's a powerful tool for quickly understanding complex multivariate data.

set.seed(123)
ggcorrmat(
  data = condo_resale[, 5:23],  
          matrix.type = "upper",
  type = "parametric",
  tr = 0.2,
  partial = FALSE,
  k = 2L,
  sig.level = 0.05,
  conf.level = 0.95,
  bf.prior = 0.707,
  ggcorrplot.args = list(
     tl.cex = 10,
     pch.cex = 5,
     lab_size = 3
  )) + 
  ggplot2::theme(
    axis.text.x = ggplot2::element_text(
      margin = ggplot2::margin(t = 0.15, r = 0.15, b = 0.15, l = 0.15, unit = "cm")
    )
  )

Reflection

Let’s breakdown the code chunk above !

  • data = condo_resale[, 5:23]: This specifies the data to be used for the correlation matrix. It selects columns 5 to 23 from the condo_resale data frame.

  • matrix.type = "upper": This argument specifies that only the upper triangle of the correlation matrix should be displayed.

  • type = "parametric": This specifies the type of correlation coefficient to be computed. In this case, it's a parametric correlation.

  • tr = 0.2: This is the transparency level for the correlation matrix.

  • partial = FALSE: This indicates that partial correlations should not be computed.

  • k = 2L: This specifies the number of decimal places to be used when displaying the correlation coefficients.

  • sig.level = 0.05: This sets the significance level for the correlation coefficients.

  • conf.level = 0.95: This sets the confidence level for the correlation coefficients.

  • bf.prior = 0.707: This sets the prior for the Bayes factor computation.

  • ggcorrplot.args = list(tl.cex = 10, pch.cex = 5, lab_size = 3): These are additional arguments passed to the ggcorrplot function, which controls the size of the text labels, points, and label size.

  • ggplot2::theme(axis.text.x = ggplot2::element_text(margin = ggplot2::margin(t = 0.15, r = 0.15, b = 0.15, l = 0.15, unit = "cm"))): This is a theme setting from the ggplot2 package that adjusts the margin around the x-axis text.

5.3 Building a Hedonic Pricing Model Using Multiple Linear Regression Method

Now, we will build a hedonic pricing model of SELLING_PRICE using multiple linear regression method that we explored in previous section.

condo.mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE    + 
                  PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                  PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + 
                  PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + 
                  PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
                  PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                data=condo_resale.sf)

tab_model(condo.mlr, show.fstat = TRUE,
  show.aic = TRUE,show.aicc = TRUE)
  Dependent variable
Predictors Estimates CI p
(Intercept) 481728.40 243504.91 – 719951.90 <0.001
AREA SQM 12708.32 11983.32 – 13433.33 <0.001
AGE -24440.82 -29861.15 – -19020.48 <0.001
PROX CBD -78669.78 -91948.06 – -65391.50 <0.001
PROX CHILDCARE -351617.91 -566353.20 – -136882.62 0.001
PROX ELDERLYCARE 171029.42 88423.78 – 253635.05 <0.001
PROX URA GROWTH AREA 38474.53 13907.81 – 63041.26 0.002
PROX HAWKER MARKET 23746.10 -33729.46 – 81221.66 0.418
PROX KINDERGARTEN 147468.99 -14697.53 – 309635.51 0.075
PROX MRT -314599.68 -428271.67 – -200927.69 <0.001
PROX PARK 563280.50 432730.10 – 693830.90 <0.001
PROX PRIMARY SCH 180186.08 52212.74 – 308159.42 0.006
PROX TOP PRIMARY SCH 2280.04 -37757.88 – 42317.95 0.911
PROX SHOPPING MALL -206604.06 -290641.86 – -122566.25 <0.001
PROX SUPERMARKET -44991.80 -196200.15 – 106216.54 0.560
PROX BUS STOP 683121.35 411722.09 – 954520.61 <0.001
NO Of UNITS -231.18 -405.83 – -56.53 0.010
FAMILY FRIENDLY 140340.77 48103.40 – 232578.14 0.003
FREEHOLD 359913.01 263360.67 – 456465.35 <0.001
Observations 1436
R2 / R2 adjusted 0.652 / 0.647
AIC 42970.175
AICc 42970.769

With reference to the table above, it is clear that not all the independent variables are statistically significant (i.e. some variables resulted in p-value > 0.05). We will revised the model by removing those variables which are not statistically significant.

condo.mlr1 <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                   PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                   PROX_URA_GROWTH_AREA + PROX_MRT  + PROX_PARK + 
                   PROX_PRIMARY_SCH + PROX_SHOPPING_MALL    + PROX_BUS_STOP + 
                   NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
                 data=condo_resale.sf)

tab_model(condo.mlr, show.fstat = TRUE,
  show.aic = TRUE,show.aicc = TRUE)
  Dependent variable
Predictors Estimates CI p
(Intercept) 481728.40 243504.91 – 719951.90 <0.001
AREA SQM 12708.32 11983.32 – 13433.33 <0.001
AGE -24440.82 -29861.15 – -19020.48 <0.001
PROX CBD -78669.78 -91948.06 – -65391.50 <0.001
PROX CHILDCARE -351617.91 -566353.20 – -136882.62 0.001
PROX ELDERLYCARE 171029.42 88423.78 – 253635.05 <0.001
PROX URA GROWTH AREA 38474.53 13907.81 – 63041.26 0.002
PROX HAWKER MARKET 23746.10 -33729.46 – 81221.66 0.418
PROX KINDERGARTEN 147468.99 -14697.53 – 309635.51 0.075
PROX MRT -314599.68 -428271.67 – -200927.69 <0.001
PROX PARK 563280.50 432730.10 – 693830.90 <0.001
PROX PRIMARY SCH 180186.08 52212.74 – 308159.42 0.006
PROX TOP PRIMARY SCH 2280.04 -37757.88 – 42317.95 0.911
PROX SHOPPING MALL -206604.06 -290641.86 – -122566.25 <0.001
PROX SUPERMARKET -44991.80 -196200.15 – 106216.54 0.560
PROX BUS STOP 683121.35 411722.09 – 954520.61 <0.001
NO Of UNITS -231.18 -405.83 – -56.53 0.010
FAMILY FRIENDLY 140340.77 48103.40 – 232578.14 0.003
FREEHOLD 359913.01 263360.67 – 456465.35 <0.001
Observations 1436
R2 / R2 adjusted 0.652 / 0.647
AIC 42970.175
AICc 42970.769 
tbl_regression(condo.mlr1,
               intercept = TRUE)
Characteristic Beta 95% CI1 p-value
(Intercept) 527,633 315,417, 739,849 <0.001
AREA_SQM 12,778 12,057, 13,498 <0.001
AGE -24,688 -30,092, -19,284 <0.001
PROX_CBD -77,131 -88,436, -65,826 <0.001
PROX_CHILDCARE -318,473 -530,250, -106,696 0.003
PROX_ELDERLYCARE 185,576 107,303, 263,849 <0.001
PROX_URA_GROWTH_AREA 39,163 16,105, 62,222 <0.001
PROX_MRT -294,745 -406,394, -183,096 <0.001
PROX_PARK 570,505 442,004, 699,006 <0.001
PROX_PRIMARY_SCH 159,856 41,698, 278,014 0.008
PROX_SHOPPING_MALL -220,947 -292,668, -149,226 <0.001
PROX_BUS_STOP 682,482 418,616, 946,348 <0.001
NO_Of_UNITS -245 -418, -73 0.005
FAMILY_FRIENDLY 146,308 54,321, 238,295 0.002
FREEHOLD 350,600 255,448, 445,752 <0.001
1 CI = Confidence Interval

5.4 Visualising Model Parameters

The function ggcoefstats() generates dot-and-whisker plots for regression models saved in a tidy data frame. The tidy data frames are prepared using parameters::model_parameters(). Additionally, if available, the model summary indices are also extracted from performance::model_performance().

mlr.p <- ggcoefstats(condo.mlr1)
mlr.p

Reflection

We can also use sort = argument to specify whether we want to sort the coefficient estimates in "ascending" or "descending" order. By default, sort = NULL is specified and no ordering is implemented.

5.5 Checking for Multicollinearity

In this section, we will explore a R package specially programmed for performing OLS regression. It is called olsrr. It provides a collection of very useful methods for building better multiple linear regression models:

  • comprehensive regression output

  • residual diagnostics

  • measures of influence

  • heteroskedasticity tests

  • collinearity diagnostics

  • model fit assessment

  • variable contribution assessment

  • variable selection procedures

Now that we have built a multiple linear regression in previous session, we will now use ols_vif_tol() of olsrr package to test if there are sign of multicollinearity.

multicol_stats <- ols_vif_tol(condo.mlr1)
tableHTML(multicol_stats)
Variables Tolerance VIF
1 AREA_SQM 0.872855423242667 1.14566510486352
2 AGE 0.707127520156393 1.41417208564989
3 PROX_CBD 0.635614652878236 1.57328028149088
4 PROX_CHILDCARE 0.306601856967953 3.26155884993391
5 PROX_ELDERLYCARE 0.659847919847265 1.51550072360836
6 PROX_URA_GROWTH_AREA 0.751031083374135 1.33150281278283
7 PROX_MRT 0.523608983366243 1.90982208435592
8 PROX_PARK 0.827926085868263 1.20783729015046
9 PROX_PRIMARY_SCH 0.452462836020451 2.21012626980661
10 PROX_SHOPPING_MALL 0.673879496684337 1.48394483720051
11 PROX_BUS_STOP 0.351411792499116 2.84566432130337
12 NO_Of_UNITS 0.690103613311802 1.44905776568972
13 FAMILY_FRIENDLY 0.724415713651706 1.38042284444535
14 FREEHOLD 0.693116329580593 1.44275925601854

Since the VIF of the independent variables are less than 10. We can safely conclude that there are no sign of multicollinearity among the independent variables.

5.6 Test for Non-Linearity

In multiple linear regression, it is important for us to test the assumption that linearity and additivity of the relationship between dependent and independent variables.

We will use ols_plot_resid_fit() of olsrr package to perform linearity assumption test.

ols_plot_resid_fit(condo.mlr1)

The figure above reveals that most of the data points are scattered around the 0 line, hence we can safely conclude that the relationships between the dependent variable and independent variables are linear.

5.7 Test for Normality Assumption

Lastly, we will use ols_plot_resid_hist() of olsrr package to perform normality assumption test.

ols_plot_resid_hist(condo.mlr1)

normality_stats <- ols_test_normality(condo.mlr1)
normality_stats
-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------

The summary table above reveals that the p-values of the four tests are way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis and infer that there is statistical evidence that the residual are not normally distributed.

5.8 Test for Spatial Autocorrelation

The hedonic model we try to build are using geographically referenced attributes, hence it is also important for us to visual the residual of the hedonic pricing model.

In order to perform spatial autocorrelation test, we need to convert condo_resale.sf from sf data frame into a SpatialPointsDataFrame.

  • First, we will export the residual of the hedonic pricing model and save it as a data frame.

  • Next, we will join the newly created data frame with condo_resale.sf object.

  • Next, we will convert condo_resale.res.sf from simple feature object into a SpatialPointsDataFrame because spdep package can only process sp conformed spatial data objects.

mlr.output <- as.data.frame(condo.mlr1$residuals)
condo_resale.res.sf <- cbind(condo_resale.sf, 
                        condo.mlr1$residuals) %>%
rename(`MLR_RES` = `condo.mlr1.residuals`)
condo_resale.sp <- as_Spatial(condo_resale.res.sf)
condo_resale.sp
class       : SpatialPointsDataFrame 
features    : 1436 
extent      : 14940.85, 43352.45, 24765.67, 48382.81  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs 
variables   : 23
names       : POSTCODE, SELLING_PRICE, AREA_SQM, AGE,    PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN,    PROX_MRT,   PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ... 
min values  :    18965,        540000,       34,   0, 0.386916393,    0.004927023,      0.054508623,          0.214539508,        0.051817113,       0.004927023, 0.052779424, 0.029064164,      0.077106132,          0.077106132,                  0, ... 
max values  :   828833,       1.8e+07,      619,  37, 19.18042832,     3.46572633,      3.949157205,           9.15540001,        5.374348075,       2.229045366,  3.48037319,  2.16104919,      3.928989144,          6.748192062,        3.477433767, ...

Next, we will use tmap package to display the distribution of the residuals on an interactive map.

tm_shape(mpsz_svy21)+
  tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha = 0.4) +
tm_shape(condo_resale.res.sf) +  
  tm_dots(col = "MLR_RES",
          palette = "plasma",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))

The figure above seems to indicate that there is sign of spatial autocorrelation. However, to prove that our observation is indeed true, the Moran’s I test will be performed.

First, we will compute the distance-based weight matrix by using dnearneigh() function of spdep.

nb <- dnearneigh(coordinates(condo_resale.sp), 0, 1500, longlat = FALSE)
nb_summary <- summary(nb)
nb_summary
Neighbour list object:
Number of regions: 1436 
Number of nonzero links: 66266 
Percentage nonzero weights: 3.213526 
Average number of links: 46.14624 
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24 
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47 
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44 
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12 
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33 
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84 
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6 
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104 
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1 
105 106 107 108 109 110 112 116 125 
  1   5   9   2   1   3   1   1   1 
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links

Next, nb2listw() of spdep packge will be used to convert the output neighbours lists (i.e. nb) into a spatial weights.

nb_lw <- nb2listw(nb, style = 'W')
summary(nb_lw)
Characteristics of weights list object:
Neighbour list object:
Number of regions: 1436 
Number of nonzero links: 66266 
Percentage nonzero weights: 3.213526 
Average number of links: 46.14624 
10 disjoint connected subgraphs
Link number distribution:

  1   3   5   7   9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24 
  3   3   9   4   3  15  10  19  17  45  19   5  14  29  19   6  35  45  18  47 
 25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44 
 16  43  22  26  21  11   9  23  22  13  16  25  21  37  16  18   8  21   4  12 
 45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 
  8  36  18  14  14  43  11  12   8  13  12  13   4   5   6  12  11  20  29  33 
 65  66  67  68  69  70  71  72  73  74  75  76  77  78  79  80  81  82  83  84 
 15  20  10  14  15  15  11  16  12  10   8  19  12  14   9   8   4  13  11   6 
 85  86  87  88  89  90  91  92  93  94  95  96  97  98  99 100 101 102 103 104 
  4   9   4   4   4   6   2  16   9   4   5   9   3   9   4   2   1   2   1   1 
105 106 107 108 109 110 112 116 125 
  1   5   9   2   1   3   1   1   1 
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links

Weights style: W 
Weights constants summary:
     n      nn   S0       S1       S2
W 1436 2062096 1436 94.81916 5798.341

Next, lm.morantest() of spdep package will be used to perform Moran’s I test for residual spatial autocorrelation

lm.morantest(condo.mlr1, nb_lw)

    Global Moran I for regression residuals

data:  
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
weights: nb_lw

Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I      Expectation         Variance 
    1.438876e-01    -5.487594e-03     3.758259e-05

The Global Moran’s I test for residual spatial autocorrelation shows that it’s p-value is less than 0.00000000000000022 which is less than the alpha value of 0.05. Hence, we will reject the null hypothesis that the residuals are randomly distributed.

Since the Observed Global Moran I = 0.1424418 which is greater than 0, we can infer than the residuals resemble cluster distribution.

6.0 Building Hedonic Pricing Model using GWmodel

After exploring the use of linear regression and multiple linear regression in previous sessions, we will now explore how to model hedonic pricing using both the fixed and adaptive bandwidth schemes.

GWR is an outgrowth of ordinary least squares regression (OLS); and adds a level of modeling sophistication by allowing the relationships between the independent and dependent variables to vary by locality. Note that the basic OLS regression model above is just a special case of the GWR model where the coefficients are constant over space. The parameters in the GWR are estimated by weighted least squares. The weighting matrix is a diagonal matrix, with each diagonal element wij being a function of the location of the observation. The role of the weight matrix is to give more value to observations that are close to i, as it is assumed that observations that are close will influence each other more than those that are far away (Tobler’s Law).

There are three major decisions to make when running a GWR: (1) the bandwidth h of the function, which determines the degree of distance decay, (2) the kernel density function assigning weights wij ,and (3) who to count as neighbors.

6.1 Computing Bandwidth

To calculate the optimal bandwidth to use in the model, bw.gwr() of GWModel package can be used, with both fixed and adapative mode. Also, There are two possible approaches can be uused to determine the stopping rule, they are: CV cross-validation approach and AIC corrected (AICc) approach. We define the stopping rule using approach argeement.

bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                     PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                     PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                     FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale.sp, 
                   approach="CV", 
                   kernel="gaussian", 
                   adaptive=FALSE, 
                   longlat=FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14 
Fixed bandwidth: 10917.26 CV score: 7.970454e+14 
Fixed bandwidth: 6749.419 CV score: 7.273273e+14 
Fixed bandwidth: 4173.553 CV score: 6.300006e+14 
Fixed bandwidth: 2581.58 CV score: 5.404958e+14 
Fixed bandwidth: 1597.687 CV score: 4.857515e+14 
Fixed bandwidth: 989.6077 CV score: 4.722431e+14 
Fixed bandwidth: 613.7939 CV score: 1.379526e+16 
Fixed bandwidth: 1221.873 CV score: 4.778717e+14 
Fixed bandwidth: 846.0596 CV score: 4.791629e+14 
Fixed bandwidth: 1078.325 CV score: 4.751406e+14 
Fixed bandwidth: 934.7772 CV score: 4.72518e+14 
Fixed bandwidth: 1023.495 CV score: 4.730305e+14 
Fixed bandwidth: 968.6643 CV score: 4.721317e+14 
Fixed bandwidth: 955.7206 CV score: 4.722072e+14 
Fixed bandwidth: 976.6639 CV score: 4.721387e+14 
Fixed bandwidth: 963.7202 CV score: 4.721484e+14 
Fixed bandwidth: 971.7199 CV score: 4.721293e+14 
Fixed bandwidth: 973.6083 CV score: 4.721309e+14 
Fixed bandwidth: 970.5527 CV score: 4.721295e+14 
Fixed bandwidth: 972.4412 CV score: 4.721296e+14 
Fixed bandwidth: 971.2741 CV score: 4.721292e+14 
Fixed bandwidth: 970.9985 CV score: 4.721293e+14 
Fixed bandwidth: 971.4443 CV score: 4.721292e+14 
Fixed bandwidth: 971.5496 CV score: 4.721293e+14 
Fixed bandwidth: 971.3793 CV score: 4.721292e+14 
Fixed bandwidth: 971.3391 CV score: 4.721292e+14 
Fixed bandwidth: 971.3143 CV score: 4.721292e+14 
Fixed bandwidth: 971.3545 CV score: 4.721292e+14 
Fixed bandwidth: 971.3296 CV score: 4.721292e+14 
Fixed bandwidth: 971.345 CV score: 4.721292e+14 
Fixed bandwidth: 971.3355 CV score: 4.721292e+14 
Fixed bandwidth: 971.3413 CV score: 4.721292e+14 
Fixed bandwidth: 971.3377 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3396 CV score: 4.721292e+14 
Fixed bandwidth: 971.3402 CV score: 4.721292e+14 
Fixed bandwidth: 971.3398 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3399 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
                     PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA + 
                     PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                     FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale.sp, 
                   approach="CV", 
                   kernel="gaussian", 
                   adaptive=TRUE, 
                   longlat=FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14 
Adaptive bandwidth: 561 CV score: 7.667364e+14 
Adaptive bandwidth: 354 CV score: 6.953454e+14 
Adaptive bandwidth: 226 CV score: 6.15223e+14 
Adaptive bandwidth: 147 CV score: 5.674373e+14 
Adaptive bandwidth: 98 CV score: 5.426745e+14 
Adaptive bandwidth: 68 CV score: 5.168117e+14 
Adaptive bandwidth: 49 CV score: 4.859631e+14 
Adaptive bandwidth: 37 CV score: 4.646518e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Adaptive bandwidth: 25 CV score: 4.430816e+14 
Adaptive bandwidth: 32 CV score: 4.505602e+14 
Adaptive bandwidth: 27 CV score: 4.462172e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14

6.2 Building Basic GWModel with Fixed and Adaptive Bandwidth

Now we can use the fixed and adaptive bandwidth values above to calibrate the gwr model using gaussian kernel (which is the default kernel density function).

gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE  + PROX_URA_GROWTH_AREA +
      PROX_MRT   + PROX_PARK + PROX_PRIMARY_SCH + 
      PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS + 
                      FAMILY_FRIENDLY + FREEHOLD, 
                       data=condo_resale.sp, 
                       bw=bw.fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)

gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + 
                            PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE + 
                            PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK + 
                            PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                          data=condo_resale.sp, bw=bw.adaptive, 
                          kernel = 'gaussian', 
                          adaptive=TRUE, 
                          longlat = FALSE)

gwr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-03-11 15:17:04.894653 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian", 
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 971.34 
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3599e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7426e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5001e+06 -1.5970e+05  3.1970e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8074e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112794435
   AREA_SQM                 21575
   AGE                     434203
   PROX_CBD               2704604
   PROX_CHILDCARE         1654086
   PROX_ELDERLYCARE      38867861
   PROX_URA_GROWTH_AREA  78515805
   PROX_MRT               3124325
   PROX_PARK             18122439
   PROX_PRIMARY_SCH       4637517
   PROX_SHOPPING_MALL     1529953
   PROX_BUS_STOP         11342209
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720745
   FREEHOLD               6073642
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3807 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6193 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71 
   Residual sum of squares: 2.534069e+14 
   R-square value:  0.8909912 
   Adjusted R-square value:  0.8430418 

   ***********************************************************************
   Program stops at: 2024-03-11 15:17:06.376575 
gwr.adaptive
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-03-11 15:17:06.377451 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian", 
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08 
   Residual sum of squares: 2.528227e+14 
   R-square value:  0.8912425 
   Adjusted R-square value:  0.8561185 

   ***********************************************************************
   Program stops at: 2024-03-11 15:17:08.020667

Based on the results, two conclusions can be made as below.

  • The AICc of the fixed-bandwidth GWR model is 42263.61 which is significantly smaller than the globel multiple linear regression model of 42967.1.

  • The AICc the adaptive-bandwidth GWR model is 41982.22 which is even smaller than the AICc of the fixed-bandwidth GWR model, which is 42263.61.

6.3 Visualisaing GWR Output

In addition to regression residuals, the output feature class table includes fields for observed and predicted y values, condition number (cond), Local R2, residuals, and explanatory variable coefficients and standard errors:

  • Condition Number: this diagnostic evaluates local collinearity. In the presence of strong local collinearity, results become unstable. Results associated with condition numbers larger than 30, may be unreliable.

  • Local R2: these values range between 0.0 and 1.0 and indicate how well the local regression model fits observed y values. Very low values indicate the local model is performing poorly. Mapping the Local R2 values to see where GWR predicts well and where it predicts poorly may provide clues about important variables that may be missing from the regression model.

  • Predicted: these are the estimated (or fitted) y values 3. computed by GWR.

  • Residuals: to obtain the residual values, the fitted y values are subtracted from the observed y values. Standardized residuals have a mean of zero and a standard deviation of 1. A cold-to-hot rendered map of standardized residuals can be produce by using these values.

  • Coefficient Standard Error: these values measure the reliability of each coefficient estimate. Confidence in those estimates are higher when standard errors are small in relation to the actual coefficient values. Large standard errors may indicate problems with local collinearity.

They are all stored in a SpatialPointsDataFrame or SpatialPolygonsDataFrame object integrated with fit.points, GWR coefficient estimates, y value, predicted values, coefficient standard errors and t-values in its “data” slot in an object called SDF of the output list.

To visualise the fields in SDF, we need to first covert it into sf data frame.

condo_resale.sf.adaptive <- st_as_sf(gwr.adaptive$SDF) %>%
  st_transform(crs=3414)
glimpse(condo_resale.sf.adaptive)
Rows: 1,436
Columns: 52
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ AREA_SQM                <dbl> 9561.892, 16576.853, 13091.861, 20730.601, 672…
$ AGE                     <dbl> -9514.634, -58185.479, -26707.386, -93308.988,…
$ PROX_CBD                <dbl> -120681.94, -149434.22, -259397.77, 2426853.66…
$ PROX_CHILDCARE          <dbl> 319266.925, 441102.177, -120116.816, 480825.28…
$ PROX_ELDERLYCARE        <dbl> -393417.795, 325188.741, 535855.806, 314783.72…
$ PROX_URA_GROWTH_AREA    <dbl> -159980.203, -142290.389, -253621.206, -267929…
$ PROX_MRT                <dbl> -299742.96, -2510522.23, -936853.28, -2039479.…
$ PROX_PARK               <dbl> -172104.47, 523379.72, 209099.85, -759153.26, …
$ PROX_PRIMARY_SCH        <dbl> 242668.03, 1106830.66, 571462.33, 3127477.21, …
$ PROX_SHOPPING_MALL      <dbl> 300881.390, -87693.378, -126732.712, -29593.34…
$ PROX_BUS_STOP           <dbl> 1210615.44, 1843587.22, 1411924.90, 7225577.51…
$ NO_Of_UNITS             <dbl> 104.8290640, -288.3441183, -9.5532945, -161.35…
$ FAMILY_FRIENDLY         <dbl> -9075.370, 310074.664, 5949.746, 1556178.531, …
$ FREEHOLD                <dbl> 303955.61, 396221.27, 168821.75, 1212515.58, 3…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…
summary(gwr.adaptive$SDF$yhat)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
  171347  1102001  1385528  1751842  1982307 13887901

We will now visualise the local R2 value as below.

tmap_mode("view")
tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "Local_R2",
          palette = c("#57bfc0","#7977f3", "#ce77b4","#f67774","#f89974", "#f8d673","#f9f777"),
          border.col = "gray20",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))
tmap_mode("plot")

6.4 Visualisaing GWR Coefficient Estimates

Next, we will visualise the coefficient estimates

tmap_mode("view")
AREA_SQM_SE <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_SE",
          palette=c("#57bfc0","#7977f3", "#ce77b4","#f67774","#f89974", "#f8d673","#f9f777"),
          border.col = "gray20",
          border.lwd = 1)

AREA_SQM_TV <- tm_shape(mpsz_svy21)+
  tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +  
  tm_dots(col = "AREA_SQM_TV",
          palette=c("#57bfc0","#7977f3", "#ce77b4","#f67774","#f89974", "#f8d673","#f9f777"),
          border.col = "gray20",
          border.lwd = 1)

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, 
             asp=1, ncol=2,
             sync = TRUE)

By URA Planning Area

tmap_mode("plot")
tm_shape(mpsz_svy21[mpsz_svy21$REGION_N=="CENTRAL REGION", ])+
  tm_polygons()+
tm_shape(condo_resale.sf.adaptive) + 
  tm_bubbles(col = "Local_R2",
            palette=c("#57bfc0","#7977f3", "#ce77b4","#f67774","#f89974", "#f8d673","#f9f777"),
            size = 0.15,
            border.col = "gray20",
            border.lwd = 1)